Problem: If $f(x)=f(2-x)$ for all $x$, then what line is necessarily an axis of symmetry of the graph of $y=f(x)$? (Give the simplest equation of this line.)
Solution: For every point $(x,y)$ on the graph of $y=f(x)$, we know $(2-x,y)$ is also on the graph of $y=f(x)$.

We have $x = 1+(x-1)$ and $2-x = 1-(x-1)$, so the geometric transformation which takes $(x,y)$ to $(2-x,y)$ is reflection across the vertical line $\boxed{x=1}$.